According to the A-theory of time, there are three monadic and irreducible temporal properties: pastness, presentness, and futurity. Among other things, this means that there’s a distinguished present — where time passes — and this distinguished present is usually understand to spread across the entire universe. A-theorists typically think that there’s a fact about what’s happening now on the planet Jupiter, for example. Everything that’s objectively present is objectively simultaneous with everything else that’s objectively present. Long ago, it was pointed out that any ordinary version of A-theory is incompatible with any ordinary interpretation of Special Relativity for an obvious reason: Special Relativity does not include a notion of absolute simultaneity and so does not include an objective present. On the orthodox understanding of Special Relativity, any two numerically distinct events that are simultaneous in one reference frame are not simultaneous in other reference frames; since there’s no objectively right reference frame, there’s no fact about which numerically distinct events take place at the same time as which other numerically distinct events.

Some A-theorists think that they’ve come up with a solution. Despite how Special Relativity is often taught and thought about, i.e., in terms of a four dimensional space-time and so on, there’s another theory — Neo-Lorentzianism — that makes many of the same predictions as Special Relativity makes, but is consistent with Special Relativity. Neo-Lorentzians argue that there are universal forces that operate on bodies in just the right way that they reproduce the effects predicted by Special Relativity; for example, these forces compress bodies along their direction of motion and slow down clocks. In published work, I’ve referred to the combination of A-theory and Neo-Lorentzianism as ANL.

One obvious problem for friends of ANL is that we have another theory, General Relativity, that supplanted Special Relativity, and that General Relativity makes different problems for A-theory than those made by Special Relativity.

Friends of ANL often tell us that they think General Relativity is a more hospitable environment for A-theory than Special Relativity is. As they point out, there are families of solutions to the Einstein Field Equations (for example, the FLRW solutions and some spatially bounded universes) that appear to have a special foliation; in turn, perhaps we can think of the objective present as one of the hypersurfaces in such a foliation. They could also point out that the “nice” relationship between the Hamiltonian formulation of General Relativity and quantum theory, so that we may have indications that whatever quantum gravity theory supplants General Relativity could end up entailing A-theory or something like it. And they could point out that there’s a nice relationship between the initial value formulation of General Relativity, global hyperbolicity, and the need for a distinguished collection of hypersurfaces.

But there’s also a well known problem that General Relativity makes for A-theory that goes back at least to Kurt Godel. Importantly, there are solutions to the Einstein Field Equations that don’t permit an interpretation in terms of any standard version of A-theory. For example, there are solutions to the Einstein Field Equations that are not globally hyperbolic. And without global hyperbolicity, it seems like ordinary A-theory couldn’t possibly be true.

I once pointed this out to someone, who then asked me what global hyperbolicity is. Since I wanted to phrase the issue in non-technical terms, I told them that global hyperbolicity is a condition that forbids time travel. But, they replied, A-theorists have long known how to make sense of time travel. So, there’s no problem, they claimed, and no reason that A-theorists need be committed to global hyperbolicity.

Today, I was reading Paul Davies’s How to Build a Time Machine, which I use in my intro course, and I came across a simple way to put the point. In chapter 3, Davies explains how, using our current best theories of fundamental physics, we can put together the plans for building a time machine, even if we currently have no way to actually carry the plans out. Davies’s time machine involves a stable wormhole. Davies is imagining that we first create a wormhole where the two mouths are located near each other in time and space. And then we place one mouth of the wormhole close to a neutron star and the other mouth far away. Due to gravitational time dilation, the two mouths are no longer located near each other in time, at least from one perspective. But this isn’t true from another perspective. Davies writes:

The gravity of the neutron star stretches time at mouth A; the clock there will run slow. What about the clock at B? Since it is some light years from the star, its rate should not be affected by the star’s gravity, so it should tick considerably faster then clock at A. But there’s a catch. Suppose we look through the wormhole from mouth A, located near the star. We then see clock B just a few meters away. So by one route the clock at B is very far from the neutron star; by another it is very near. If it is regarded as very near, then time at B should be slowed by the star’s gravity too. There should be very little difference between the clocks’ rates at A and B. So which view is right? The answer is, both. Time is, after all, relative, and the situation here is that, viewed through the wormhole, time is about the same at both ends; but, viewed across the ‘outer’ space, the time difference between A and B (clock B is ahead of A) is substantial. If you now jump through the wormhole from A to B, you will jump back ten years into the past (Davies 2001, pg. 90).

So, here’s the problem for the A-theorist. If we try to find some foliation of space-time where we can identify the hypersurfaces included in that foliation as instants of time (including the present) then we find that there is no one hypersurface that is (for example) present but not past. There will be hypersurfaces that, for example, extend through the wormhole but are also numerically identical to some past instant. The wormhole alters the topology of space-time in a really radical way that shouldn’t be possible on A-theory.

For example, presentism — the most popular version of A-theory — maintains that only the present exists. Past instants no longer exist. But if a present instant is not distinct from a past instant, then the past hasn’t really passed away.

Here’s a closely related surgical procedure to see what’s going on here. Start with a globally hyperbolic space-time. Now cut a “slit” at one location and another slit at some other location. Identify the points on both sides of the slit, so that trajectories that travel into the slit from one side exit at the other. The resulting space-time is obviously not globally hyperbolic and may allow for (for example) closed time-like curves.

Since such a space-time allows for closed time-like curves, it’s incompatible with any ordinary version of A-theory in yet another way. Explaining precisely why is a bit tricky if we suppose, as many A-theorists do, that endurantism is true. Begin, then, by supposing that perdurantism is true and that I traverse a closed time-like curve. In that case, my present temporal part, one of my past temporal parts, and one of my future temporal parts are all numerically identical. Any perdurantists who accept ordinary A-theory would regard that as impossible. On the other hand, endurantists say that I am one and the same object over time, so they wouldn’t have a problem saying that there’s something in the future or the past that’s numerically identical with my present self. But they’re still not going to be okay with saying that my present-self is just the same thing as my future-self or past-self. For the endurantist, one and the same object has distinct properties indexed to different times — even though I-today am the same object as I-tomorrow, the properties that I have today are not the properties that I have tomorrow. Hence, endurantists who accept ordinary A-theory shouldn’t be friendly to closed time-like curves either.